Show that $n$ can be written in the form $n=c_{0}+c_{1}d+...+c_{k}d^{k}$

Asked Oct 09 '20 at 13:04

Active Oct 09 '20 at 13:04

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I day ago, i've been solving some induction exercises from my textbook. But when i saw this, it seems a bit tricky and i couldn't come up with a solution. I hope someone can give clarity for this. Thanks!

Let $n$, $d$ positive integers and assume $1<d<n$. Show that $n$ can be written in the form $$n=c_{0}+c_{1}d+...+c_{k}d^{k}$$ with $0≤c_{i}<d$, and that these integers $c_{i}$ are uniquely determined.

I first proved the uniqueness of $n$ using Euclidean-Algorithm with $n=qd+c_{0}$ what i am troubling now, is how will i conclude this finding to show that $n$ can be written on that form?

asked Oct 09 '20 at 13:04

EquDox

1

Division algoritm? Notice that the problem asks you the well-posedness of writing $n$ in base $d$ – FormulaWriter Oct 09 '20 at 13:09
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Is $c_{r+1}$ uniquely determined from the proof? – EquDox Oct 09 '20 at 13:23
which proof do you mean? Anyway, consider this hint: set $d=10$. How would you proceed? – FormulaWriter Oct 09 '20 at 13:30
https://math.stackexchange.com/questions/1125578/showing-that-every-positive-integer-can-be-represented-in-this-form, Btw thanks for the hint! – EquDox Oct 09 '20 at 13:48

Show that $n$ can be written in the form $n=c_{0}+c_{1}d+...+c_{k}d^{k}$

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